Pairs of selfadjoint operators and their invariants

D. Alpay, I. Gohberg

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

A trace formula is proved for pairs of selfadjoint operators that are close to each other in a certain sense. An important role is played by a function analytic in the open upper half-plane and with positive imaginary part there. This function, called the characteristic function of the pair, coincides withKreĭn’s Q-function in the case where the selfadjoint operators are canonical extensions of a common simple and closed Hermitian operator. Special emphasis is given to the finite-dimensional case. Relationships with Kreĭn’s spectral shift function are also considered. Finally, the case of canonical differential expressions is discussed briefly. In this case, the function N may be chosen to be the Weyl function of the canonical differential expression.

Original languageEnglish
Pages (from-to)59-104
Number of pages46
JournalSt. Petersburg Mathematical Journal
Volume16
Issue number1
DOIs
StatePublished - 1 Jan 2004

Keywords

  • Kreĭn’s spectral shift function
  • The Q-function associated with a symmetric operator
  • The Weyl function

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